The Stäckel systems and algebraic curves

Abstract

We show how the Abel–Jacobi map provides all the principal properties of an ample family of integrable mechanical systems associated to hyperelliptic curves. We prove that derivative of the Abel–Jacobi map is just the Stäckel matrix, which determines n-orthogonal curvilinear coordinate systems in a flat space. The Lax pairs, r-matrix algebras and explicit form of the flat coordinates are constructed. An application of the Weierstrass reduction theory allows us to construct several flat coordinate systems on a common hyperelliptic curve and to connect among themselves different integrable systems on a single phase space.